26 September 2007

Fun with images! (Mostly about math tattoos.)

Stokes' theorem graffiti. In a bathroom. Only in Camberville. (I would have said "only in Cambridge", but this particular graffito is actually in Somerville. Diesel Cafe, to be precise; there was a summer where I spent a lot of time there doing math, so I'm not surprised.)

I can think of others. An ex-girlfriend had an infinity symbol tattooed on her (although I believe that was more of a literary thing than a scientific one). A friend of mine has

¬ (p ∧ ¬ p))

tattooed on his leg. A friend of a friend in college had the Taylor series for sine on her arm. It's written out, as

except with exactly enough terms to go once around her arm; it's not in the more compact form

.

I don't have any tattoos; I can't think of anything I'd want permanently inked on me. But if I were to get a tattoo, the equation

would be up there. (I had it written on my wall in college.) I've heard it called the "computer scientist's identity"; it's an analytic version of the fact that every positive integer has a unique binary expansion. The left-hand side expands to

(1+z1)(1+z2)(1+z4)(1+z8)...

and you can imagine expanding that out; then you get something like

1 + z1 + z2 + z2+1 + z4 + z4+1 + z4+2 + z4+2+1 + ...

where each sum of distinct powers of two appears as an exponent exactly once. The right-hand side is the geometric series

1/(1-z) = 1 + z + z2 + z3 + ...

and in order for these two expressions to be equal, the coefficient of zn in each of them has to be equal. All the coefficients on the right-hand side are 1; thus the coefficients on the left-hand side must be too. That means each nonnegative integer appears exactly once as an exponent, i. e. as a sum of distinct powers of two. The fact that one can disguise this combinatorial fact as a fact about functions from C to C -- and then apply the methods of complex analysis, although they're not incredibly useful in this particular case -- is a fascinating example of the interplay between the discrete and the continuous.